Hochberg Correction

[Nadra999]

… I am trying to use the Hochberg correction on some data I have.

The unadjusted ps look like this:

Brids p=0.0001

Fish p=0.0009

Mammals = 0.1402

And after Hochberg like this using SAS PROC MULTEST:

Brids p=0.0004

Fish p=0.0018

Mammals = 0.1402

The values in the correction just don’t seem right…

For reference:

http://www.statisticshowto.com/benjamini-hochberg-procedure/

How to Run the Benjamini–Hochberg procedure

1. Put the individual p-values in ascending order.
2. Assign ranks to the p-values. For example, the smallest has a rank of 1, the second smallest has a rank of 2.
3. Calculate each individual p-value’s Benjamini-Hochberg critical value, using the formula (i/m)Q, where:
   • i = the individual p-value’s rank,
   • m = total number of tests,
   • Q = the false discovery rate (a percentage, chosen by you).
4. Compare your original p-values to the critical B-H from Step 3; find the largest p value that is smaller than the critical value.

Step 1:

So with our data they are already in order:

Brids p=0.0001

Fish p=0.0009

Mammals = 0.1402

Step 2:

Birds Rank 1

Fish Rank 2

Mammals Rank 3

Step 3:

Birds:

i=1

m=3

Q=0.05

(i/m)Q=(1/3)0.05=0.016

I am not sure what the Q value is in SAS and cannot find documentation. But, if you work back from the values provided by SAS 1/3x=0.016, then you get a Q of 0.0012. That seems like a strange value…

also the Q would change….

For fish if 2/3Q=0.0018 then Q=0.0027

For mammals 3/3Q=0.1402, then Q=0.1402

[ballardw · Accepted solution]

I don't see a Q in the documentation under Hochberg for Multtest:

The Hochberg-adjusted p-values are defined in reverse order of the step-down Bonferroni:

shows that the adjusted p-values for the highest ranked (I=m) = to the original p-value, hence 0.1402 for mammals in your case.

For m=2, it would be the minimum of p-value for i=3 (0.1402), (3-2+1)(0.0009) {the second ranked p-value], or min(0.1402, 2*0.0009)=0.0018.

So the adjustment used by Multtest does calculate the results you show from the documentation but there is no assumed Q. It appears that the Benjamini–Hochberg is a different adjustment than the Hochberg that Multtest uses. I have to admit that I am always uncomfortable with a process including something like: Q = the false discovery rate (a percentage, chosen by you).

What would you say is the false discovery rate from your data?